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Some Properties of the Intervals MML Identifier:MEASURE6.
"... for this paper. The following propositions are true: (1) There exists a function F from N into [:N, N:] such that F is onetoone and domF = N and rngF = [:N, N:]. (2) For every function F from N into R such that F is nonnegative holds 0 R ≤ ∑F. (3) Let F be a function from N into R and x be an ext ..."
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for this paper. The following propositions are true: (1) There exists a function F from N into [:N, N:] such that F is onetoone and domF = N and rngF = [:N, N:]. (2) For every function F from N into R such that F is nonnegative holds 0 R ≤ ∑F. (3) Let F be a function from N into R and x be an extended real number. Suppose there exists a natural number n such that x ≤ F(n) and F is nonnegative. Then x ≤ ∑F. (8) 1 For all extended real numbers x, y such that x is a real number holds (y − x)+x = y and (y+x) − x = y. (10) 2 For all extended real numbers x, y, z such that z ∈ R and y < x holds (z+x)−(z+y) = x−y. (11) For all extended real numbers x, y, z such that z ∈ R and x ≤ y holds z + x ≤ z + y and x+z ≤ y+z and x − z ≤ y − z. (12) For all extended real numbers x, y, z such that z ∈ R and x < y holds z + x < z + y and x+z < y+z and x − z < y − z. Let x be a real number. The functor R(x) yielding an extended real number is defined as follows: (Def. 1) R(x) = x. We now state a number of propositions: (13) For all real numbers x, y holds x ≤ y iff R(x) ≤ R(y). (14) For all real numbers x, y holds x < y iff R(x) < R(y). (15) For all extended real numbers x, y, z such that x < y and y < z holds y is a real number. (16) Let x, y, z be extended real numbers. Suppose x is a real number and z is a real number and x ≤ y and y ≤ z. Then y is a real number. (17) For all extended real numbers x, y, z such that x is a real number and x ≤ y and y < z holds y is a real number.
Some Properties of the Intervals MML Identifier: MEASURE6.
"... and notation for this paper. The scheme FunctXD YD concerns a non empty set A, a non empty set B, and a binary predicate P, and states that: There exists a function F from A into B such that for every element x of A holds P[x,F(x)] provided the following condition is satisfied: • For every element x ..."
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and notation for this paper. The scheme FunctXD YD concerns a non empty set A, a non empty set B, and a binary predicate P, and states that: There exists a function F from A into B such that for every element x of A holds P[x,F(x)] provided the following condition is satisfied: • For every element x of A there exists an element y of B such that P[x,y]. Let X, Y be non empty sets. Note that Y X is non empty. We now state a number of propositions: (1) There exists a function F from ¦ into [: ¦ , ¦:] such that F is onetoone and domF = ¦ and rng F = [: ¦ , ¦:]. (2) For every function F from ¦ into ¤ such that F is nonnegative holds 0 § ≤ � F. (3) Let F be a function from ¦ into ¤ and let x be a Real number. Suppose there exists a natural number n such that x ≤ F(n) and F is nonnegative. Then x ≤ � F. (4) For every Real number x such that there exists a Real number y such that y < x holds x � = −∞. (5) For every Real number x such that there exists a Real number y such that x < y holds x � = +∞. (6) For all Real numbers x, y holds x ≤ y iff x < y or x = y. (7) Let x, y be Real numbers and let p, q be real numbers. If x = p and y = q, then p ≤ q iff x ≤ y. (8) For all Real numbers x, y such that x is a real number holds (y−x)+x = y and (y + x) − x = y. (9) For all Real numbers x, y such that x ∈ ¤ holds x + y = y + x.
Some Properties of the Intervals MML Identifier: MEASURE6.
"... this paper. The following propositions are true: (1) There exists a function F from N into [: N, N :] such that F is onetoone and domF = N and rngF = [: N, N :]. (2) For every function F from N into R such that F is nonnegative holds 0 F ..."
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this paper. The following propositions are true: (1) There exists a function F from N into [: N, N :] such that F is onetoone and domF = N and rngF = [: N, N :]. (2) For every function F from N into R such that F is nonnegative holds 0 F
Properties of subsets
 Journal of Formalized Mathematics
, 1989
"... Summary. The text includes theorems concerning properties of subsets, and some operations on sets. The functions yielding improper subsets of a set, i.e. the empty set and the set itself are introduced. Functions and predicates introduced for sets are redefined. Some theorems about enumerated sets a ..."
Partial Functions
"... this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following rules ..."
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Cited by 492 (10 self)
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this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following
The fundamental properties of natural numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
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Cited by 688 (73 self)
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Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1
RatioBased Decisions and the Quantitative Analysis of cDNA Microarray Images
, 1997
"... Gene expression can be quantitatively analyzed by hybridizing fluortagged mRNA to targets on a cDNA microarray. Comparison of gene expression levels arising from cohybridized samples is achieved by taking ratios of average expression levels for individual genes. A novel method of image segmentation ..."
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Cited by 366 (28 self)
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segmentation is provided to identify cDNA target sites and a hypothesis test and confidence interval is developed to quantify the significance of observed differences in expression ratios. In particular, the probability density of the ratio and the maximumlikelihood estimator for the distribution are derived
Vascular permeability factor (vascular endothelial growth factor gene is expressed differentially in normal tissues, macrophages and tumors
 MA BiA Cell. 3:211
, 1992
"... Persistent microvascular hyperpermeability to plasma proteins even after the cessation of injury is a characteristic but poorly understood feature of normal wound healing. It results in extravasation of fibrinogen that clots to form fibrin, which serves as a provisional matrix and promotes angiogene ..."
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Cited by 367 (9 self)
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identified in healing splitthickness guinea pig and rat punch biopsy skin wounds by their capacity to extravasate circulating macromolecular tracers (colloidal carbon, fluoresceinated dextran). Vascular permeability was maximal at 23 d, but persisted as late as 7 d after wounding. Leaky vessels were found
Binary operations
 Journal of Formalized Mathematics
, 1989
"... Summary. In this paper we define binary and unary operations on domains. We also define the following predicates concerning the operations:... is commutative,... is associative,... is the unity of..., and... is distributive wrt.... A number of schemes useful in justifying the existence of the operat ..."
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Cited by 363 (6 self)
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of the operations are proved. MML Identifier:BINOP_1. WWW:http://mizar.org/JFM/Vol1/binop_1.html The articles [4], [3], [5], [6], [1], and [2] provide the notation and terminology for this paper. Let f be a function and let a, b be sets. The functor f(a, b) yielding a set is defined by: (Def. 1) f(a, b) = f(〈a, b
MML Identifier: PENCIL 4.
"... Summary. In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of kpencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definitio ..."
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Summary. In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of kpencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definition presented in the paper [8] by Naumowicz and Pra˙zmowski. For all spaces defined, we state the conditions under which they are not degenerated to a single line.
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